A power series $f(z)$ which converges in $\mathbb {D}=\{|z|<1\}$ maps the radii $[0,\zeta )$ onto paths $\Gamma (\zeta )$, $\zeta \in \mathbb {T}=\partial \mathbb {D}$. These are studied under several aspects in the case of the special lacunary series $f(z)=z+z^2+z^4+z^8+\ldots$. First, the $\Gamma (\zeta )$ are considered as random functions on the probability space $(\mathbb {T},\mathscr {B},\mathrm {mes}/2\pi )$, where $\mathscr {B}$ is the $\sigma$-algebra of Borel sets and $\mathrm {mes}$ the Lebesgue measure. Then analytical properties of the $\Gamma (\zeta )$ are discussed which hold on subsets $A$ of $\mathbb {T}$ with Hausdorff dimension 1 in spite of $\mathrm {mes}{A}=0$. Furthermore, estimates of the derivative of $f$ and of the arc length of sections of the $\Gamma (\zeta )$ are given. Finally, these results are used to derive connections between the distribution of critical points of $f$ and the overall behaviour of the paths.